The DOI matches its Crossref record; 19 of the 19 references it cites were found in a registry or an index.
Hyperbolic Groups
Overview
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“Following prior work, we first advocate for using hyperbolic spaces which provably model tree-like structures better than Euclidean geometry.”
From Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (2018) · cites this paper as Gromov, 1987
In the citing paper
“For instance, the Euclidean space R n for n ≥ 2 is not δ -hyperbolic for any δ ≥ 0 , while the Poincar´ e ball D n is log(1+ √ 2) -hyperbolic.”
From Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (2018) · cites this paper as Gromov, 1987
In the citing paper
“Theorem : For any δ > 0 , any δ -hyperbolic metric space ( X,d X ) and any set of points x 1 , ..., x n ∈ X , there exists a finite weighted tree ( T, d T ) and an embedding f : T → X such that for all i, j ,”
From Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (2018) · cites this paper as Gromov, 1987
In the citing paper
“This exponential growth property enables hyperbolic spaces to embed any weighted tree while almost preserving their metric 3 (Gromov, 1987; Bowditch, 2006; Sarkar, 2011).”
From Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (2018) · cites this paper as Gromov, 1987
Citations
Cites
Lyndon, R. C., & Schupp, P. E. (2001) · Classics in mathematics · doi:10.1007/978-3-642-61896-3 · Cited by 1,026
Gromov, M. (1983) · Journal of Differential Geometry · doi:10.4310/jdg/1214509283 · Cited by 946
Self-citationFree subgroups in linear groups
Tits, J. (1972) · Journal of Algebra · doi:10.1016/0021-8693(72)90058-0 · Cited by 825
Manifolds of Nonpositive Curvature
Ballmann, W., Gromov, M., & Schroeder, V. (1985) · Progress in mathematics · doi:10.1007/978-1-4684-9159-3 · Cited by 784
Self-citationKlingenberg, W. (1978) · Grundlehren der mathematischen Wissenschaften · doi:10.1007/978-3-642-61881-9 · Cited by 500
Markov Partitions for Axiom A Diffeomorphisms
Bowen, R. (1970) · American Journal of Mathematics · doi:10.2307/2373370 · Cited by 484
Geometry and Topology of Manifolds
Boden, H. U., Hambleton, I., Nicas, A., & Park, B. (2005) · American Mathematical Society eBooks · doi:10.1090/fic/047 · Cited by 417
Valuations, Trees, and Degenerations of Hyperbolic Structures, I
Morgan, J. W., & Shalen, P. B. (1984) · Annals of Mathematics · doi:10.2307/1971082 · Cited by 340
Groups Generated by reflections and aspherical manifolds not covered by Euclidean space
Davis, M. W. (1983) · Annals of Mathematics · doi:10.2307/2007079 · Cited by 337
On the existence and irreducibility of certain series of representations
Kostant, B. (1969) · Bulletin of the American Mathematical Society · doi:10.1090/s0002-9904-1969-12235-4 · Cited by 271
Research works printed but not linked: 14
- A.D. Alexandrov, A theorem on triangles in a metric space and some of its applications, Trudy Math. Inst. Steks. 38 (1951), 5–23.
- N. Bourbaki, XXXIV, Groups et algébres de Lie, Part 2, Hermann, Paris, 1968.
- Yu. Burago and V. Zalgaller, Geometric Inequalities, Egrebn. der Math., Springer-Verlag, to appear.
- A. Dress, Trees, extensions of metric spaces and the cohomological dimension of certain groups, Preprint, Bielefeld.
- W.J. Floyd, Group completions and Kleinian groups, Ph.D. Thesis, Princeton, 1978.
- M. Gromov, Hyperbolic manifolds, groups and actions, In. Riemannian surfaces and related topics, Ann. Math. Studies 97 (1981), 183–215.
- M. Gromov, Hyperbolic manifolds, Volume and bounded cohomology, Publ. Math. IHES 56 (1983), 213–307.
- M. Gromov, Hyperbolic manifolds, Infinite groups as geometric objects, Proc. ICM Waszawa, Vol 1 (1984), 385–391.
- D. Kazdan, On a connection between the dual space of a group with the structure of it’s closed subgroups, Funct. An. and Applications, 1:1 (1967), 71–74.
- R.C. Lyndon and P.E. Schupp, Combinatorial group theory, Springer-Verlag (1977).
- G. Margulis, The isometry of closed manifolds of constant negative curvature with the same fundamental group, Dokl. Ak. Nauk. SSSR 192 (1970), 736–737.
- P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symmetriques, preprint.
- D. Sullivan, Quasiconformal homeomorphisms and dynamics, II, Preprint IHES, 1982.
- J.L. Tits, On buildings and their applications, Proc. I.C.M. 1974, pp. 209–221, Vancouver.
Cited by
Sorted by citations
Zomorodian, A., & Carlsson, G. (2005) · Discrete & Computational Geometry · doi:10.1007/s00454-004-1146-y · Cited by 1,591
Convex polytopes, Coxeter orbifolds and torus actions
Davis, M. W., & Januszkiewicz, T. (1991) · Duke Mathematical Journal · doi:10.1215/s0012-7094-91-06217-4 · Cited by 646
Carnot-Carathéodory spaces seen from within
Gromov, M. (1996) · Birkhäuser Basel eBooks · doi:10.1007/978-3-0348-9210-0_2 · Cited by 638
Self-citationGeometry of the Space of Phylogenetic Trees
Billera, L. J., Holmes, S., & Vogtmann, K. (2001) · Advances in Applied Mathematics · doi:10.1006/aama.2001.0759 · Cited by 575
Coverage in sensor networks via persistent homology
de Silva, V., & Ghrist, R. (2007) · Algebraic & Geometric Topology · doi:10.2140/agt.2007.7.339 · Cited by 422
Metadata
- Authors
- Gromov, M.
- Corresponding author
- Gromov, M.
- Venue
- Mathematical Sciences Research Institute Publications
- Publisher
- Springer New York
- Published
- 1987
- Pages
- 75-263
- ISSN
- 0940-4740 (ISSN-L)
- OpenAlex
- W4235484205
- Cited by
- 1,817
- References
- 19
- Citations by year
- 2026: 41 · 2025: 106 · 2024: 94 · 2023: 92 · 2022: 87 · 2021: 90 · 2020: 72 · 2019: 69 · 2018: 72 · 2017: 82 · 2016: 79 · 2015: 75 · 2014: 68 · 2013: 66 · 2012: 54
- FWCI
- 19.75
- Type
- book chapter
- Language
- en
- Open access
- closed
- Locations
- Mathematical Sciences Research Institute publications (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Mathematics › Geometry and Topology › Mathematics and Applications
- Keywords
- hyperbolic groups, Gromov hyperbolicity, geometric group theory, Cayley graphs, word metric, hyperbolic geometry
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
